# 12.2.1 - Hypothesis Testing

In testing the statistical significance of the relationship between two quantitative variables we will use the five step hypothesis testing procedure:

1. Check assumptions and write hypotheses

In order to use Pearson's $$r$$ both variables must be quantitative and the relationship between $$x$$ and $$y$$ must be linear

Research Question Is the correlation in the population different from 0? Is the correlation in the population positive? Is the correlation in the population negative?
Null Hypothesis, $$H_{0}$$ $$\rho=0$$ $$\rho= 0$$ $$\rho = 0$$
Alternative Hypothesis, $$H_{a}$$ $$\rho \neq 0$$ $$\rho > 0$$ $$\rho< 0$$
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
2. Calculate the test statistic

Use Minitab Express to compute $$r$$

3. Determine the p-value

Minitab Express will give you the p-value for a two-tailed test (i.e., $$H_a: \rho \neq 0$$). If you are conducting a one-tailed test you will need to divide the p-value in the output by 2.

4. Make a decision

If $$p \leq \alpha$$ reject the null hypothesis, there is evidence of a relationship in the population.

If $$p>\alpha$$ fail to reject the null hypothesis, there is not evidence of a relationship in the population.

5. State a "real world" conclusion.

Based on your decision in Step 4, write a conclusion in terms of the original research question.

## Optional: t Test Statistic Section

If you are conducting a test by hand, a $$t$$ test statistic is computed in step 2 using the following formula:

$$t=\dfrac{r- \rho_{0}}{\sqrt{\dfrac{1-r^2}{n-2}}}$$

In step 3, a $$t$$ distribution with $$df=n-2$$ is used to obtain the p-value.